tag:blogger.com,1999:blog-8231784566931768362.post8372252867792595154..comments2023-09-09T08:21:55.454-04:00Comments on MathNotations: Putting the Gold 'Bach' into Primes! An Investigation...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-8231784566931768362.post-49693858327608629372007-09-25T21:41:00.000-04:002007-09-25T21:41:00.000-04:00More about The Number Devil.I used it at the end o...More about <I>The Number Devil</I>.<BR/><BR/>I used it at the end of the year last year with my 5th graders. It was that last week of school when you can't really do anything, but I still wanted them to be engaged and doing stuff. Each day during math time, I read them a chapter and then we did some work with the concepts in that chapter. We didn't make it through the whole book (by design, the later chapters get a bit beyond what my 5th graders were ready for). They really enjoyed the quick-tempered number devil. Other topics explored: primes, Pascal's triangle, infinite series. Lots more too, but the book is at school and I'm at home. I highly recommend this book. I bought it for myself, but immediately saw the applications in the classroom. Many of my kids didn't realize they were still 'doing math' when we read and did activities with the book.<BR/><BR/>I'll definitely get back to you next week after the investigation with my advanced math section (7th graders).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-1431799850915132202007-09-25T12:27:00.000-04:002007-09-25T12:27:00.000-04:00Thanks for another great blog. This morning I was ...Thanks for another great blog. This morning I was re-reading it with the intention of commending your emphasis on the process and on asking questions, but I see you've just preceded me in pointing this out :).<BR/><BR/>I believe the results we get in math, in science, or elsewhere for that matter, depend a whole lot on the process followed and the questions asked. Glad to see you're using math as a vehicle to expose students to this. <BR/><BR/>MarcAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-62814654848452636552007-09-25T11:10:00.000-04:002007-09-25T11:10:00.000-04:00thanks, mathmom, for the excellent review/recommen...thanks, mathmom, for the excellent review/recommendation!<BR/>makes me want to run to the library, being cheap and all that!<BR/>dave<BR/><BR/>Another point about this investigation. While the focus has been on primes, I hope my readers recognize that I had a few other objectives here. Exposing students to the nature of math research -- the whole notion of the process: posing questions, tackling a problem, collecting data, looking for patterns/relationships, making conjectures, generalizing/abstracting, attempting a proof or posing other questions that may lead in a different direction,...<BR/>I've always believed that this process is similar to the Scientific Method and should be explained as such to mathematics students. And the ost important personal quality for mathematical research? I suggest persistence in the face of frustration! Just ask Prof. Wiles, or, better yet, his wife!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-5211813873368264542007-09-25T10:33:00.000-04:002007-09-25T10:33:00.000-04:00The Number Devil -- this is a great book, and I wo...The Number Devil -- this is a great book, and I would say good for all ages (including adults). It has large print, color illustrations, and a fiction story line that makes it appealing even to fairly young kids (and a great choice for young gifted kids) but the mathematical content is great for middle/high school level kids, and fun for curious adults as well, especially those who may not have been exposed to all the "fun" math stuff in the past.mathmomhttps://www.blogger.com/profile/05869925405540832241noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-15158821441438014492007-09-25T09:01:00.000-04:002007-09-25T09:01:00.000-04:00Arithmetic density is just density--I added the ad...Arithmetic density is just density--I added the adjective to distinguish it from log density further down. Take an increasing sequence a_k, and its counting function a(x) = #{ a_k | a_k ≤ x}. The counting function for the primes is π(x), for example. Then, the density is lim_{n→∞} a(n)/n.<BR/><BR/>So, the density of the even numbers is 1/2, as approximately n/2 of the positive integers ≤ n are even. π(n)/n ≈ 1/log n, so the density is zero.<BR/><BR/>Log density is the limit of log a(n) / n. That the squares have log density ½ makes the results that every prime ≡ 1 (mod 4) is a sum of two squares, every number not of the form 4^k (8n + 7) the sum of three, and every number be the sum of 4 plausible. Yet, Log density is unaffected by multiplying the entire sequence by a constant. What sort of results would you have for the sequence of numbers like 2n²? That's why I said 'modified' above.<BR/><BR/>This type of measure doesn't sum the way areas and volumes sum, exactly. The integers have log density 1; so do the primes. So do the sums of two squares. They're different sets.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-951993542920348642007-09-25T06:32:00.000-04:002007-09-25T06:32:00.000-04:00lynx--The Number Devil sounds engaging. Tell us mo...lynx--<BR/>The Number Devil sounds engaging. Tell us more about it, its intended age range (kids of all ages?)and a brief description. I found it at Amazon but I haven't seen this book before. <BR/><BR/>As far as turning rich engaging mathematics into full-blown classroom investigations, well, that's why I write this blog! Come back often and, if you use it in class, let me know how students reacted and how you needed to modify it for them. Thanks!<BR/><BR/>eric--<BR/>What can I say. You are our resident expert who brings so much more to what I write. Arithmetic density was a new phrase for me, a bit different from density per se. Your wealth of knowledge is awesome.<BR/><BR/>timro21--<BR/>Nice website. I may add that to my blogroll and recommend the list of unsolved problems to the educators who stop and visit here. Thanks for sharing. I hope you enjoyed my little student activity. I delight in bringing the mysteries of primes and number theory to a larger audience.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-45432475361245969072007-09-25T05:38:00.000-04:002007-09-25T05:38:00.000-04:00Feel free to use any of the problems at my site......Feel free to use any of the problems at my site...<BR/>http://www.unsolvedproblems.org/<BR/><BR/>TimUnknownhttps://www.blogger.com/profile/06030021311779507460noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-51289030098296396202007-09-25T01:39:00.000-04:002007-09-25T01:39:00.000-04:00Ah, Lynx. Vinogradov's theorem: every sufficientl...Ah, Lynx. Vinogradov's theorem: every sufficiently large odd number is the sum of three primes. MathWorld points out that the best value of "sufficiently large is about 3.33×10^{43000}.<BR/><BR/>There's also Chen's Theorem that every sufficiently large even number is expressible as one or both of p+q or p+qr, with p, q, r prime.<BR/><BR/>The lesson for students should be the importance of the 'sufficiently large' clause; it shows up in many places in mathematics. Consider Waring's problem: for each positive integer n≥2, there is a number g(n) such that every integer is the sum of at most g(n) nth-powers. every integer is the sum of at most 4 squares, at most 9 cubers, and at most 19 4th powers.<BR/><BR/>The hard problem is to find G(n) such that every sufficiently large integer is the sum of at most G(n) nth-powers. It turns out that G(3)≤7. It may be less than 7.<BR/><BR/>The other lesson is to teach the idea of arithmetic density. There are more than one way to measure the size of a set. The even and the odd numbers both have density ½. The primes have density 0; so do the squares. But they are all infinite sets. So, what's the important measure of size for these sets? Both are important in different circumstances.<BR/><BR/>And then there's logarithmic density. The primes have log density 1. The squares have log density ½. It turns out that you expect a (modified) Goldbach or Waring-type result for any set with positive log density.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-6729006019233055082007-09-24T21:25:00.000-04:002007-09-24T21:25:00.000-04:00One way to extend it is to explore the numbers tha...One way to extend it is to explore the numbers that are the sum of three primes.<BR/><BR/>In the past, I read sections of <I>The Number Devil</I> to my students and we did some similar work. It wasn't as much of an investigation, but they were still interested. We did the three primes I mentioned above as well.<BR/><BR/>I think I'll try this with one of my math sections next week. Thanks for reminding me of it and presenting it as more of an investigation.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-77082613187587822722007-09-24T15:27:00.000-04:002007-09-24T15:27:00.000-04:00Great suggestions, mathmom. Your wording is clear ...Great suggestions, mathmom. Your wording is clear and will get them started in the right direction more quickly.<BR/>However, I've worked with some middle school math groups and the discussion that might result from intentionally leaving out details with guidance by the instructor can be of value. Yes, some will jump to the pattern of odds you mentioned but others will challenge that with with the 4 = 2+2 example and why 8 = 1+7 or 10 = 1+9 is missing from the list. Sometimes I intentionally leave it vague like this and make it part of the challenge. Try it with your own sons or some of the older students and see what happens. This is particularly effective when one has mixed ability groupings. Students will correct each other and the end result will be fine. Let me know!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-26784446398592884392007-09-24T13:25:00.000-04:002007-09-24T13:25:00.000-04:00Dave, this investigation looks great, as always. :...Dave, this investigation looks great, as always. :) <BR/><BR/>One issue I have though is that I don't think there's enough "leading" at the beginning to get them to come up with the kind of conjecture that you want. Based on your examples, I think most of my students would "conjecture" that all even numbers can be represented as the sum of two odd numbers. And oh, so much easier to prove. ;-)<BR/><BR/>I would add something like:<BR/><B>Today we are going to examine the different ways numbers may be represented as the sum of two (or more) prime numbers. Remember that 1 is not a prime number, and that 2 is the only even prime.</B><BR/><BR/>In addition to wanting to try this with my middle schoolers, I have a pair of 9yos I think I'd like to give this to as something to pull out and work on over time when they finish their other math work ahead of the rest of the class. They've both already shown a lot of interest in prime numbers and trying to find patterns related to them.<BR/><BR/>With my middle schoolers, I'm currently following a trail from Gauss sums to triangle and other figural numbers, to handshake problems, so I won't get to this soon, but I'll try to come back and tell you how it goes when I do try it. Also if I give it to the younger guys, I'll let you know what they think.mathmomhttps://www.blogger.com/profile/05869925405540832241noreply@blogger.com